SLAC-PUB-15129 QCD Analysis of the Scale-Invariance of Jets Andrew J. Larkoski * SLAC National Accelerator Laboratory, Menlo Park, CA 94025 (Dated: July 9, 2012) Abstract Studying the substructure of jets has become a powerful tool for event discrimination and for studying QCD. Typically, jet substructure studies rely on Monte Carlo simulation for vetting their usefulness; however, when possible, it is also important to compute observables with analytic meth- ods. Here, we present a global next-to-leading-log resummation of the angular correlation function which measures the contribution to the mass of a jet from constituents that are within an angle R with respect to one another. For a scale-invariant jet, the angular correlation function should scale as a power of R. Deviations from this behavior can be traced to the breaking of scale invariance in QCD. To do the resummation, we use soft-collinear effective theory relying on the recent proof of factorization of jet observables at e + e - colliders. Non-trivial requirements of factorization of the angular correlation function are discussed. The calculation is compared to Monte Carlo parton shower and next-to-leading order results. The different calculations are important in distinct phase space regions and exhibit that jets in QCD are, to very good approximation, scale invariant over a wide dynamical range. PACS numbers: 12.38.Bx,12.38.Cy,13.66.Bc,13.87.Fh * [email protected]1 Submitted to Physical Review D Work supported by US Department of Energy contract DE-AC02-76SF00515. arXiv:1207.1437v1
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SLAC-PUB-15129
QCD Analysis of the Scale-Invariance of Jets
Andrew J. Larkoski∗
SLAC National Accelerator Laboratory, Menlo Park, CA 94025
(Dated: July 9, 2012)
Abstract
Studying the substructure of jets has become a powerful tool for event discrimination and for
studying QCD. Typically, jet substructure studies rely on Monte Carlo simulation for vetting their
usefulness; however, when possible, it is also important to compute observables with analytic meth-
ods. Here, we present a global next-to-leading-log resummation of the angular correlation function
which measures the contribution to the mass of a jet from constituents that are within an angle R
with respect to one another. For a scale-invariant jet, the angular correlation function should scale
as a power of R. Deviations from this behavior can be traced to the breaking of scale invariance
in QCD. To do the resummation, we use soft-collinear effective theory relying on the recent proof
of factorization of jet observables at e+e− colliders. Non-trivial requirements of factorization of
the angular correlation function are discussed. The calculation is compared to Monte Carlo parton
shower and next-to-leading order results. The different calculations are important in distinct phase
space regions and exhibit that jets in QCD are, to very good approximation, scale invariant over
The current success of the Large Hadron Collider (LHC), its high center of mass energies,
its significant delivered integrated luminosity and its high-precision experiments has ushered
in a new era of particle physics. Particles and jets with significant transverse boosts are now
being copiously produced. An entire field of studying the substructure of highly boosted
jets has grown up out of the study of these objects and many methods have been proposed
to study QCD. In addition, procedures for discriminating QCD jets from jets initiated by
heavy particle decays have been introduced and new measurements of these methods are
being completed [1, 2]. To understand these methods in detail, most analyses have relied
on Monte Carlo simulation as the basis of study. However, Monte Carlo simulations have
limitations, and, where possible, it is vital to also compute the observables to higher orders
in QCD so as to have another handle on their behavior.
An important contribution to this effort of computing jet observables is resummation
of large logarithms that arise in fixed-order perturbation theory. Jets are objects that
are typically dominated by soft or collinear emissions and so it is necessary to resum the
logarithms that exist for an accurate prediction of an observable. Very recently, groups
have computed resummed contributions to light jet masses at hadron colliders [3] and N-
subjettiness [4, 5] in color-singlet jets at the LHC [6]. Ref. [6] in particular relied on the
factorization of color singlet processes at hadron colliders to reinterpret results from e+e−
colliders. Computing the resummed contribution to generic observables at hadron colliders is
made more difficult by the color flow throughout the collision which can destroy factorization.
To avoid discussion of these issues, here we will only consider jet observables at e+e− colliders.
In this paper, we will discuss the resummation of the angular correlation function introduced
in [7] using soft collinear effective theory (SCET) [8–11].
The angular correlation function G(R) was defined in [7] as
G(R) =∑
i 6=jp⊥ip⊥j∆R
2ijΘ(R−∆Rij) , (1)
for studying the substructure of jets at the LHC. ∆Rij is the boost-invariant angle between
particles i and j, the sum runs over all constituents of a jet and Θ is the Heaviside theta
function. The angular correlation function has distinct properties for scaleless jets versus jets
with at least one heavy mass scale. In particular, any structure in the angular correlation
2
function should be distributed roughly as RD, where D is a constant, for a scaleless jet.
It was shown that by exploiting the different behavior of the angular distribution of hard
structure in QCD jets versus jets initiated by heavy particle decay, an efficient tagging
algorithm could be defined.
Ref. [12] continued studying the properties of the angular correlation function, focusing
on average properties of QCD jets. It was shown through simple calculations that, for QCD
jets, the angular correlation function averaged over an ensemble of jets should approximately
scale as
〈G(R)〉 ' R2 , (2)
where the angle brackets are defined by
〈G(R)〉 =1
Njets
Njets∑
i=1
G(R)i . (3)
Deviations from R2 are due to the running coupling and higher order effects. The introduc-
tion of an ensemble averaged angular correlation function allows for a rigorous definition of
the dimension of a QCD jet which is also infrared and collinear (IRC) safe. This dimension
is defined to be the average angular structure function 〈∆G〉 and is the power to which the
average angular correlation function scales with R:
〈∆G(R)〉 ≡ d log〈G(R)〉d logR
. (4)
For QCD jets, 〈∆G〉 ∼ 2. In [12], it was also shown that the scaling of non-perturbative
physics in R is distinctly different, and this was used to determine the average energy density
of the underlying event.
Here, we will continue the work of [12] and compute the average angular structure function
by resummation within the context of SCET. Our analysis is only truly appropriate at e+e−
colliders, but we expect that the largest effect in going to hadron colliders is the contribution
of the underlying event. For this calculation, we introduce generalized correlation functions
Ga(R) parametrized by an index a:
Ga(R) =1
2E2J
∑
i 6=jEiEj sin θij tana−1
θij2
Θ(R− θij) (5)
The form of the angular correlation function is similar to jet angularity [13, 14]. However,
for our purposes here, we choose to index the parameter a such that the angular correlation
3
function is IRC safe for all a > 0. In the small angle limit, this reduces to Eq. 1 with a = 2
(up to normalization). The parameter a allows for a study of the behavior of the angular
correlation function with angular scales weighted differently. Analogously to the angular
structure function, we define a generalized average angular structure function
〈∆Ga〉 ≡d log〈Ga〉d logR
. (6)
The calculation and interpretation of average angular structure function will be the focus of
this paper.
In Sec. II, we discuss the factorization of jet observables in SCET and the computation
of the angular correlation function including global next-to-leading-log (NLL) contributions
for jets defined by a kT -type algorithm [15, 16]. The existence of a factorization theorem for
the angular correlation function is non-trivial. We will discuss the consistency conditions
that the angular correlation function satisfies for factorization. We will also briefly discuss
how the results obtained here can be used in a calculation of the angular correlation function
at the LHC. In Sec. III we compare the SCET calculation to a next-to-leading-order (NLO)
calculation of the angular correlation function. Resummation and fixed-order corrections
affect different parts of distributions and so the differences between the resummed calculation
and the fixed-order result give some sense as to the importance of these effects. This analysis
leads to Sec. IV, were we present a comparison between the SCET calculation and the output
of parton shower Monte Carlo. We observe significant differences between SCET and Monte
Carlo, but higher fixed order effects are substantial. We discuss some of the uncertainties in
the parton shower studying the effect of the evolution variable on the value of the angular
structure function. Finally, we present our conclusions in Sec. V.
II. SCET CALCULATION
SCET is an effective theory of QCD in which all modes of QCD are integrated out except
those corresponding to soft or collinear modes. Collinear and soft modes are defined by their
scaling with power counting parameter λ:
collinear ∼ (λ2, 1, λ) ,
soft ∼ (λ2, λ2, λ2) ,
4
which is the scaling of the +, − and transverse components of the momenta, respectively.
λ is a parameter that is defined for a particular process or observable; for example, for
computing the distribution of jet masses, λ ∼ mJ/p⊥J � 1. The fact that λ� 1 allows for
a systematic expansion in powers of λ. Higher order terms in λ are power suppressed (much
like the subleading terms in the twist expansion).
For an event shape observable O that factorizes, the cross section can be written in the
schematic form:
dσ
dO = H(µ)
[∏
ni
Jni(O;µ)
]⊗ S(O;µ) , (7)
where H(µ) is the hard function, which matches the full QCD result at a scale µ, J(µ;ni,O)
is the jet function for the contribution to the observable O from ni-collinear modes and
S(µ;O) is the soft function for the contribution to the observable O from the soft modes.
⊗ represents a convolution between the jet and soft functions. All functions depend on the
factorization scale µ.
Factorization of jet observables in SCET was first exhibited in [17, 18]. Ref. [18] computed
individual jet angularities to NLL in e+e− collisions. It was shown that factorization of the
cross section for jet observables in e+e− → N jets has the form
dσ
dO1 · · · dOM= H(n1, . . . , nN ;µ)
[M∏
i=1
Jni(Oi;µ)
]⊗ Sn1···nN
(O1, . . . ,OM ;µ)N∏
j=M+1
J(µ) ,(8)
where M ≤ N of the jet observables Oi have been measured. Jet directions are denoted by
ni and J(Oi;µ) is the jet function for a jet in which the observable Oi has been measured
and J(µ) is a jet function for a jet which has not been measured. We will refer to these as
the measured and unmeasured jet functions, respectively. A similar nomenclature will be
used for the soft functions. Jet algorithm dependence and jet energies have been suppressed.
An important point from [18] is that factorization requires that the jets be well-separated;
namely, that
tij =tan
ψij
2
tan R0
2
� 1 , (9)
where ψij the is angle between any pair of jets i, j and R0 is the jet algorithm radius. We will
assume that this condition is met in the following and leave any discussion of subtleties to
[18]. A non-trivial requirement of the factorization is the independence of the cross section
on the factorization scale µ. This requirement leads to a constraint that the sum of the
5
anomalous dimensions of the hard, jet and soft functions is zero. We will show that this
holds for the angular correlation function.
We will use the results of the factorization theorem proven in [17, 18] to compute the
distribution of the angular correlation function from Eq. 5. In particular, we are interested
in the ensemble average of the angular structure function as defined in Eq. 6. Note that this
observable is independent of any normalization factor of the angular correlation function;
thus, with the goal of computing the average angular structure function, it is consistent to
ignore factors that are independent of Ga and the angular resolution parameter R. Thus,
for the purposes of this paper, we can ignore the overall factors in the factorized form of
the cross section of the hard function and the unmeasured jet functions. In this case, the
factorized form of the cross section becomes
dσ
dGa1 · · · dGaM= C(µ)
[M∏
i=1
Jni(Gai;µ)
]⊗ Sn1···nN
(Ga1, . . . ,GaM ;µ) , (10)
where C(µ) is independent of Ga and the resolution parameter R.
In this section, we present a calculation of the jet and soft functions for the angular
correlation function for jets defined by a kT algorithm. We first argue that the angular
correlation function is computable in SCET and relate its form at NLO to the form of jet
angularity at NLO. This comparison will allow us to relate the calculation of the angular
correlation function to the work in [18]. We then present a calculation of the measured jet
and soft functions of the angular correlation function. From these results, we can determine
the anomalous dimensions of the jet and soft functions and will show the consistency of the
factorization relies on a non-trivial cancellation of dependence on the angular resolution R
between the jet and soft functions. We can then resum up to the next-to-leading logs of
the jet and soft functions by the renormalization group. Note that we do not attempt to
resum non-global logs [19] that arise due to the non-trivial phase space constraints of the
jet algorithm or the angular correlation function. From the resummed expression of the
angular correlation function, we find the ensemble average and compute the average angular
structure function numerically.
It should be stressed that non-global logarithms are ignored in this study. The angular
correlation function for a jet requires several phase space constraints; the jet algorithm, soft
jet vetoes, the resolution parameter R, etc. These provide numerous sources for non-global
logarithms which cannot be resummed analytically. The study of non-global logarithms in
6
QCD cross sections is a subtle and evolving story. For recent work in this direction, especially
in the context of non-global logarithms from jet clustering see, for example, [20–25]. It is
outside the scope of this paper to discuss non-global logarithms further.
A. Factorization of the Angular Correlation Function
Factorization of jet observables requires that soft modes only resolve the entire jet and
not individual collinear modes contributing to the jet. Angularity τa is a one-parameter
family of observables defined as [13, 14]
τa =1
2EJ
∑
i∈Je−ηi(1−a)p⊥i , (11)
where J is the jet, p⊥i is the momentum of particle i transverse to the jet axis and ηi is the
rapidity of particle i with respect to the jet axis:
ηi = − log tanθi2. (12)
Angularity is IRC safe for a < 2. The separation of soft and collinear modes in angularity
is simple to show. To leading power in λ,
τa =1
2EJ
∑
C∈Je−ηC(1−a)p⊥C +
1
2EJ
∑
S∈Je−ηS(1−a)p⊥S
= τCa + τSa , (13)
where C and S represent the collinear and soft modes, respectively. Note that the soft
modes do not affect the location of the jet center to leading power in λ. Factorization of
angularities exists only for a < 1 due to the presence of logarithms of rapidity; however,
recently it was shown that these logarithms can be controlled [26, 27]. We will show that
angularity and the angular correlation function have similarities which will allow us to use
many of the results from [18] here.
To justify the use of SCET for computing the angular correlation function, we must first
show that the angular correlation function does not mix soft and collinear modes. This
argument was presented in [12] (based on arguments from [28]), but we present it here for
completeness. In terms of soft and collinear modes, the angular correlation function can be
7
expressed as
Ga(R) =1
2E2J
∑
i 6=jEiEj sin θij tana−1
θij2
Θ(R− θij)
=1
2E2J
∑
i,j∈CEiEj sin θij tana−1
θij2
Θ(R− θij)
+1
2E2J
∑
i,j∈SEiEj sin θij tana−1
θij2
Θ(R− θij)
+1
2E2J
∑
C,S
ECES sin θCS tana−1θCS2
Θ(R− θCS) . (14)
Note that, to NLO, there is no soft-soft correlation contribution to the angular correlation
function because such a term would require the radiation of two soft gluons which first occurs
at NNLO. To accuracy of the leading power in λ, we can exchange the collinear modes with
the jet itself in the collinear-soft term. Explicitly,
θCS = θJS +O(λ) , (15)
as the angle of the soft modes with respect to the jet center scales as θJS ∼ 1. Appropriate
for NLO or NLL, the angular correlation function can be written as
Ga(R) =1
2E2J
∑
i,j∈CEiEj sin θij tana−1
θij2
Θ(R− θij)
+1
2EJ
∑
S
ES sin θJS tana−1θJS2
Θ(R− θJS) . (16)
Thus, the collinear and soft modes are decoupled to leading power and so the angular
correlation function is factorizable, and hence computable, in SCET.
To NLO, a jet is composed of at most two particles, so the form of many observables
simplifies substantially at this order. The form of the angular correlation function from Eq. 5
was chosen so as to be similar in form to angularity. The contribution to the angularity and
the angular correlation function from collinear modes is distinct. The measured jet functions
will need to be recomputed for the angular correlation function. However, the contributions
to the angularity and the angular correlation function from soft modes are simply related:
GSa (R) =ES2EJ
sin θJS tana−1θJS2
Θ(R− θJS) = τS2−aΘ(R− θJS) . (17)
This observation will allow us to recycle the soft function calculation for angularity for the
angular correlation function.
8
An important point to note here is that the scaling of the angle between collinear modes
i and j goes like θij ∼ λ. Thus, to leading power, the angular correlation function for the
collinear-collinear contribution can be written as
GCCa =1
2E2J
∑
i,j∈CEiEj sin θij tana−1
θij2
Θ(R− θij)
=1
E2J
∑
i,j∈CEiEj tana
θij2
Θ(R− θij) . (18)
We will use this form of the collinear-collinear contribution to the angular correlation func-
tion for computing the measured jet functions.
B. Measured Jet Functions
The leading power contribution to the measured jet functions at NLO comes from two
collinear particles which are clustered in the jet and can be computed from cutting one-
loop SCET diagrams. The phase space integrals can be extended over the entire range of
momentum for the collinear particles in the jet as long as the contribution from the zero
momentum bin is subtracted [29]. In particular, we consider a jet with light cone momentum
l = (l+, ω, 0) which splits to two collinear particles with light cone momenta q = (q+, q−,q⊥)
and l− q = (l+ − q+, ω − q−,−q⊥). The zero-bin subtraction term can be determined from
the measured jet function by taking the scaling q ∼ λ2. We will refer to contribution to the
jet function that does not include the zero-bin subtraction as the naıve contribution.
To compute the measured jet function, we will need to enforce phase space cuts from
the jet algorithm and the observable. We will compute the jet function for a kT -type jet
algorithm as defined by a jet radius R0. At NLO, all kT algorithms are the same and two
particles are clustered in the jet if their angular separation is less than R0. This leads to
the phase space constraint
ΘkT = Θ
(cosR0 −
q · (l− q)
|q|√
(l− q)2
)= Θ
(tan2 R0
2− q+ω2
q−(ω − q−)2
), (19)
where on the right, the leading scaling behavior was kept. The jet algorithm constraint for
the zero-bin subtraction term is then
Θ(0)kT
= Θ
(tan2 R0
2− q+
q−
). (20)
9
The phase space constraints for the angular correlation function are more subtle. The
δ-function which constrains a jet to have angular correlation function Ga, δR = δ(Ga − Ga),is
δR = δ
(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)), (21)
where R is the resolution parameter of the angular correlation function. For a kT -type jet at
NLO, the angular correlation function vanishes if R > R0; thus, we will assume that R < R0
in the following. This δ-function can be decomposed depending on the value of Θ-function
as
δR = δ
(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
))
= δ(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)
+ δ (Ga) Θ
(q+ω2
q−(ω − q−)2− tan2 R
2
). (22)
The δ-function for the zero-bin subtraction term is found by taking q ∼ λ2:
δ(0)R = δ
(Ga − ω−1(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+
q−
)+ δ (Ga) Θ
(q+
q−− tan2 R
2
). (23)
1. Measured Quark Jet Function
The naıve contribution to the measured quark jet function can be computed in dimen-
sional regularization from the diagrams shown in Fig. 1:
Jqω(Ga) = g2µ2εCF
∫dl+
2π
1
(l+)2
∫ddq
(2π)d
(4l+
q−+ (d− 2)
l+ − q+ω − q−
)
×2πδ(q+q− − q2⊥
)Θ(q−)Θ(q+)2πδ
(l+ − q+ − q2⊥
ω − q−)
×Θ(ω − q−)Θ(l+ − q+)Θ
(tan2 R0
2− q+ω2
q−(ω − q−)2
)
×[δ(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)
+ δ (Ga) Θ
(q+ω2
q−(ω − q−)2− tan2 R
2
)]. (24)
We take d = 4− 2ε. The coefficient to the δ(Ga) term can be found by integrating over Ga.The terms that remain are +-distributions, which integrate to zero. The zero-bin subtraction
10
(B)(A) (D)(C)(A) (A)
Figure 4: Diagrams contributing to the quark jet function. (A) and (B) Wilson line emission
diagrams; (C) and (D) QCD-like diagrams. The momentum assignments are the same as in Fig. 3.
The zero bin of particle 2 is given by the replacement q ! l � q.
For all the jet algorithms we consider, the zero-bin subtractions of the unmeasured jet
functions are scaleless integrals.12 However, for the measured jet functions, the zero-bin
subtractions give nonzero contributions that are needed for the consistency of the e↵ective
theory.
In the case of a measured jet, in addition to the phase space restrictions we also demand
that the jet contributes to the angularity by an amount ⌧a with the use of the delta function
�R = �(⌧a � ⌧a), which is given in terms of q and l by
�R ⌘ �R(q, l+) = �
✓⌧a �
1
!(! � q�)a/2(l+ � q+)1�a/2 � 1
!(q�)a/2(q+)1�a/2
◆. (4.4)
In the zero-bin subtraction of particle 1, the on-shell conditions can be used to write the
corresponding zero-bin �-function as
�(0)R = �
✓⌧a �
1
!(q�)a/2(q+)1�a/2
◆, (4.5)
(and for particle 2 with q ! l � q).
4.2 Quark Jet Function
The diagrams corresponding to the quark jet function are shown in Fig. 4. The fully
inclusive quark jet function is defined as
Zd4x eil·x h0|�a↵
n,!(x)�b�n,!(0) |0i ⌘ �ab
✓n/
2
◆↵�Jq!(l+) , (4.6)
and has been computed to NLO (see, e.g., [75, 76]) and to NNLO [77]. Below we compute
the quark jet function at NLO with phase space cuts for the jet algorithm for both the
measured jet, Jq!(⌧a), and the unmeasured jet, Jq
!. As discussed above, we will find that
the only nonzero contributions come from cuts through the loop when both cut particles
are inside the jet.
12Note that algorithms do exist that give nonzero zero-bin contributions to unmeasured jet functions [32].
– 32 –
FIG. 1. SCET Feynman diagrams contributing to the quark jet function.
term follows from taking the scaling limit q ∼ λ2 of the naıve jet function above:
Jq(0)ω (Ga) = 4g2µ2εCF
∫dl+
2π
1
l+
∫ddq
(2π)d1
q−2πδ
(q+q− − q2⊥
)Θ(q−)Θ(q+)
×2πδ(l+ − q+
)Θ(l+ − q+)Θ
(tan2 R0
2− q+
q−
)
×[δ(Ga − ω−1(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+
q−
)
+ δ (Ga) Θ
(q+
q−− tan2 R
2
)]. (25)
The term proportional to δ(Ga) is scaleless and integrates to zero in pure dimensional regu-
lation.
Employing a MS scheme, we find the measured quark jet function for kT -type jet algo-
rithms of Ga to be
Jqω(Ga) = Jqω(Ga)− Jq(0)ω (Ga) =αsCF
2π
[(a
a− 1
1
ε2+
3
2
1
ε+
a
a− 1
log µ2
ω2
ε+
1
εlog
tan2 R2
tan2 R0
2
)δ(Ga)
− 2
a− 1
1
ε
(Θ(Ga)Ga
)
+
]+ Jqω(Ga, ε0) , (26)
where Jqω(Ga, ε0) consists of terms that are finite as ε → 0. These terms are presented in
Appendix A. The definition of the +-distribution is also given in Appendix A. Note that
the 1/ε terms for the angular correlation function are the same as those for angularity from
[18] with a → 2 − a plus an additional term of the logarithm of the ratio of scales; the
resolution scale R and the jet radius R0. This term contributes to the anomalous dimension
of the jet function. In principle, these logarithms could be attempted to be resummed.
However, note that the resolution scale R can never practically be parametrically smaller
than the jet radius R0, so these logarithms never become large. Thus, we will not worry
about resumming these logarithms.
11
Figure 5: Diagrams contributing to the gluon jet function. (A) sunset and (B) tadpole gluon
loops; (C) ghost loop; (D) sunset and (E) tadpole collinear quark loops; (F) and (G) Wilson line
emission loops. Diagrams (F) and (G) each have mirror diagrams (not shown). The momentum
assignments are the same as in Fig. 3.
Without inserting any additional constraints, this integral is scaleless and zero in dimen-
sional regularization. Therefore, in the absence of phase-space restrictions, the naıve inte-
gral Eq. (4.19) gives the standard (inclusive) gluon jet function
Jg!(l+)
2⇡!=↵s
4⇡µ2✏(!l+)�1�✏
TRNf
✓4
3+
20
9✏
◆� CA
✓4
✏+
11
3+
✓67
9� ⇡2
◆✏
◆�, (4.21)
in the MS scheme. The measured and unmeasured jet functions are obtained by inserting
⇥alg�R and ⇥alg, respectively, into Eqs. (4.19) and (4.20).
4.3.1 Measured Gluon Jet
The naive contribution to the measured gluon jet can be written as
Jg!(⌧a) =
↵s
2⇡
1
�(1 � ✏)
✓4⇡µ2
!2
◆✏1
1 � a2
✓1
⌧a
◆1+ 2✏2�aZ 1
0dx (xa�1 + (1 � x)a�1)
2✏2�a (4.22)
⇥TRNf
✓1 � 2
1 � ✏x(1 � x)
◆� CA
✓2 � 1
x(1 � x)� x(1 � x)
◆�⇥alg(x) ,
where x ⌘ q�/!. This gives
Jg!(⌧a) =
↵s
2⇡
1
�(1 � ✏)
4⇡µ2
!2 tan2 R2
!✏�(⌧a)
"CA
✓1
✏2+
11
6
1
✏
◆� 2
3✏TRNf
#+↵s
2⇡Jg
alg(⌧a) ,
(4.23)
where, as for the quark jet function, the finite distributions Jgalg(⌧a) di↵er among the
algorithms we consider. They are given in Appendix A.
The zero-bin result is
Jg(0)! (⌧a) =
↵sCA
⇡
1
�(1 � ✏)
4⇡µ2 tan2(1�a) R
2
!2
!✏✓1
⌧a
◆1+2✏ 1
(1 � a)✏. (4.24)
– 36 –
FIG. 2. SCET Feynman diagrams contributing to the gluon jet function. Diagrams (F) and (G)
have mirrored counterparts which are not shown.
2. Measured Gluon Jet Function
The naıve contribution to the measured gluon jet function can be computed from the
diagrams shown in Fig. 2:
Jgω(Ga) = 2g2µ2ε
∫dl+
2π
1
l+
∫ddq
(2π)d1
ω − q−2πδ(q+q− − q2⊥
)2πδ
(l+ − q+ − q2⊥
ω − q−)
×{nFTR
(1− 2
1− εq+q−
ωl+
)− CA
(2− ω
q−− ω
ω − q− −q+q−
ωl+
)}
×Θ(q−)Θ(q+)Θ(ω − q−)Θ(l+ − q+)Θ
(tan2 R0
2− q+ω2
q−(ω − q−)2
)
×[δ(Ga − ωa−2(ω − q−)1−a(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+ω2
q−(ω − q−)2
)
+ δ (Ga) Θ
(q+ω2
q−(ω − q−)2− tan2 R
2
)]. (27)
The coefficient of the δ(Ga) term can be found by integrating over Ga. The terms that
remain are +-distributions, which integrate to zero. The zero-bin subtraction term follows
from taking the scaling limit l − q ∼ q ∼ λ2 of the naıve jet function above:
Jg(0)ω (Ga) = 4g2µ2εCA
∫dl+
2π
1
l+
∫ddq
(2π)d1
q−2πδ
(q+q− − q2⊥
)Θ(q−)Θ(q+)
×2πδ(l+ − q+
)Θ(l+ − q+)Θ
(tan2 R0
2− q+
q−
)
×[δ(Ga − ω−1(q−)1−a/2(q+)a/2
)Θ
(tan2 R
2− q+
q−
)
+ δ (Ga) Θ
(q+
q−− tan2 R
2
)]. (28)
The term proportional to δ(Ga) integrates to zero in pure dimensional regulation. This
zero-bin subtraction term is exactly the same up to color factors as the quark jet function
12
zero-bin subtraction.
Employing a MS scheme, we find the measured gluon jet function for the kT -type jet
algorithms of Ga to be
Jgω(Ga) = Jgω(Ga)− Jg(0)ω (Ga) =αs2π
[(CA
a
a− 1
1
ε2+β02ε
+ CAa
a− 1
log µ2
ω2
ε
+CAε
logtan2 R
2
tan2 R0
2
)δ(Ga)−
2CAε(a− 1)
(Θ(Ga)Ga
)
+
]
+ Jgω(Ga, ε0) , (29)
where Jgω(Ga, ε0) consists of terms that are finite as ε → 0. These terms are presented in
Appendix A. β0 is the coefficient of the one-loop β-function:
β0 =11
3CA −
2
3NF , (30)
with TR = 12. As with the quark jet function, the 1/ε terms are the same as those for
angularity from [18] with a→ 2− a plus an additional term of the logarithm of the ratio of
the resolution parameter R to the jet radius R0.
C. Measured Soft Function
As shown above, there is a simple relationship between the form of angularity for soft
modes and the angular correlation for soft modes. This relationship will allow us to use
the results from [18] in computing the measured soft function for the angular correlation
function. First, we consider the phase space constraints from the jet algorithm and the
angular correlation function. For the kT jet algorithm, soft radiation must be within the jet
radius R0 of the jet axis to be included:
ΘkT = Θ
(tan2 R0
2− k+
k−
). (31)
The δ-function that constrains the soft modes to contribute an amount Ga to the angular
correlation function is
δR = δ
(Ga − ω−1(k−)1−a/2(k+)a/2Θ
(tan2 R
2− k+
k−
))
= δ(Ga − ω−1(k−)1−a/2(k+)a/2
)Θ
(tan2 R
2− k+
k−
)+ δ (Ga) Θ
(k+
k−− tan2 R
2
). (32)
13
The measured soft function of a gluon emitted from lines i and j into a jet is
Smeasij (Ga) = −g2µ2εTi ·Tj
∫ddk
(2π)dni · nj
(ni · k)(nj · k)2πδ(k2)Θ(k0)ΘkT δR
= −g2µ2εTi ·Tj
∫ddk
(2π)dni · nj
(ni · k)(nj · k)2πδ(k2)Θ(k0)Θ
(tan2 R0
2− k+
k−
)
×[δ(Ga − ω−1(k−)1−a/2(k+)a/2
)Θ
(tan2 R
2− k+
k−
)
+ δ (Ga) Θ
(k+
k−− tan2 R
2
)]. (33)
Note that the integral proportional to δ(Ga) is scaleless and so vanishes in pure dimensional
regularization. Also, the integral is only non-zero if R < R0 and so the Θ-function from the
jet algorithm is redundant. Thus, we can write the soft function as
Smeasij (Ga) = −g2µ2εTi ·Tj
∫ddk
(2π)dni · nj
(ni · k)(nj · k)2πδ(k2)Θ(k0)Θ
(tan2 R
2− k+
k−
)
× δ(Ga − ω−1(k−)1−a/2(k+)a/2
). (34)
This is the same form of the measured soft function as for angularity with a jet radius equal
to R which was computed in [18]. Up to terms that are suppressed by 1/t2 from Eq. 9, the
measured soft function for jet i is
Smeas(Gia) = −αs2π
T2i
1
a− 1
{[1
ε2+
1
εlog
µ2 tan2(a−1) R2
ω2− π2
12+
1
2log2 µ
2 tan2(a−1) R2
ω2
]δ(Gia)
−2
[(1
ε+ log
µ2 tan2(a−1) R2
G2iaω2
)Θ(Gia)Gia
]
+
}, (35)
where T2i is the square of the color in the jet.
D. Anomalous Dimensions and Consistency Conditions
A non-trivial requirement of the factorization is that the physical cross section should
be independent of the factorization scale µ. A consequence of this is that the anomalous
dimensions of the hard, jet and soft functions must sum to 0. The requirement is
0 =
(γH(µ) + γunmeas
S (µ) +∑
i/∈meas
γJi(µ)
)δ(Ga) +
∑
i∈meas
(γJi(Gia;µ) + γmeas
S (Gia;µ)), (36)
where γH , γS and γJ are the anomalous dimensions of the hard, soft and jet functions. The
µ dependence must be summed over the measured and unmeasured jet and soft functions.
14
The sum of the hard, unmeasured soft and unmeasured jet anomalous dimensions to NLO
is
γH(µ) + γunmeasS (µ) +
∑
i/∈meas
γJi(µ) = −αsπ
∑
i∈meas
T2i log
µ2
ω2i tan2 R0
2
−∑
i∈meas
γi , (37)
where γi depends on the flavor of the jet:
γq =3αs2π
CF , γg =αsπ
11CA − 2NF
6=αs2πβ0 , (38)
for quark and gluon jets respectively. We will show that the measured jet and soft function
anomalous dimensions for the angular correlation function are exactly what is required to
satisfy Eq. 36.
The anomalous dimensions of the measured jet or soft functions are given by the coefficient
of the 1/ε terms from Eqs. 26, 29 and 35. The anomalous dimensions of the quark and gluon
jet functions can be written collectively as
γJi(Gia) =
[αsπT2i
(a
a− 1log
µ2
ω2i
+ logtan2 R
2
tan2 R0
2
)+ γi
]δ(Ga)− 2
αsπT2i
1
a− 1
[Θ(Ga)Ga
]
+
,
(39)
where γi is defined in Eq. 38. Note the non-trivial dependence of the anomalous dimension
on both the jet radius and the resolution parameter of the angular correlation function. The
anomalous dimension of the measured soft function for a quark or gluon jet is
γmeasS (Gia) = −αs
πT2i
1
a− 1
[δ(Ga) log
µ2 tan2(a−1) R2
ω2i
− 2
(Θ(Ga)Ga
)
+
]. (40)
As mentioned earlier, jet angularity is not factorizable for a = 1 and here we see that the
anomalous dimensions of the angular correlation jet and soft functions become meaningless
for a = 1, signaling a breakdown of factorization. For the angular correlation function, we
are most interested in a = 2, so we will not consider this issue further here.
Summing over the measured jet and soft function anomalous dimensions, we find
∑
i∈meas
(γJi(Gia;µ) + γmeas
S (Gia;µ))
=
(αsπ
∑
i∈meas
T2i log
µ2
ω2i tan2 R0
2
+∑
i∈meas
γi
)δ(Ga) (41)
Note that there is a non-trivial cancellation of the angular correlation function resolution
parameter R between the jet and soft functions. This contribution exactly cancels that from
the hard and unmeasured jet and soft functions in Eq. 37, consistent with the factorization
requirement.
15
E. Resummation and Averaging
To proceed with the resummation to NLL of the jet and soft functions, we will make a
few observations. First, as mentioned earlier, because we are ultimately interested in the
average angular structure function, we can ignore factors in the resummed cross section that
are independent of Ga or the resolution parameter R. Thus, we will not discuss nor resum the
hard function nor the unmeasured jet and soft functions. Also, we will only consider a single
measured jet in an event. This prevents a study of inter-jet correlations of the angular
correlation function, but for this paper we are most interested in the intra-jet dynamics.
Anyway, the existence of factorization of jet observables essentially trivializes correlations
between jets since it implies that correlations can only come from the soft function. From
these observations, we only need to resum the measured jet and soft functions of a single
jet.
With these considerations, we will need to compute the convolution between the measured